In this section, we show how we are able to use our vb-network to find a rank-maximal stable matching. First, we show that the Max Flow-Min Cut Theorem [1]} can be extended to a vb-network. Next, we prove that, analogous to the exponential weight case, a maximum profile closed subset of the rotation poset may be found by obtaining a minimum cut of the vb-network. Finally, we show that [2]}'s Max Flow algorithm [2]} may be adapted to work with vb-networks, and that we are able to find a rank-maximal stable matching in \(O(nm^2 \log n)\)  time using polynomially-bounded weight vectors.
